COHERENT SHEAVES ON AN ELLIPTIC CURVE

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1 COHERENT SHEAVES ON AN ELLIPTIC CURVE KRISTIAN BRÜNING, IGOR BURBAN Abstract. These are notes by Kristian Brüning of a mini-course given by Igor Burban at the summer school Derived categories in representation theory held at Tsinghua University, Beijing: August, The aim of this series of lectures is to give a classification of indecomposable coherent sheaves on an elliptic curve using a braid group action on the derived category. Contents 1. Introduction 1 2. Modules over Dedekind domains 3 3. Coherent sheaves on projective varieties 4 Functors on Coh(X) 8 Examples of coherent sheaves Coherent sheaves on an elliptic curve 10 Calabi-Yau property 11 Classification of the indecomposable coherent sheaves 16 Summary 16 References Introduction We want to study the category of coherent sheaves Coh(X) on an elliptic curve X. By definition, an elliptic curve is a smooth plain curve in P 2 of degree three (e.g. zy 2 = x 3 z 2 x), and in the affine chart z 0 it looks like: This category Coh(X) has nice properties: it is an abelian, finite dimensional hereditary category and is therefore Krull-Schmidt [Ste75, VIII,4.3]. We shall classify the indecomposable coherent sheaves on X using a braid group action on the derived category D b (Coh(X)). It yields a background for the 1

2 2 KRISTIAN BRÜNING, IGOR BURBAN study of the Hall algebra of the category Coh(X) [BS]. Before dealing with coherent sheaves on projective curves we first take a look at the affine case. Fix an algebraically closed field k. Unless otherwise stated, all rings which we shall consider are commutative and are algebras over the field k. Definition 1.1. Let I = (f 1,...,f s ) be an ideal in k[x 1,...,X n ]. The corresponding affine variety X = V (I) k n = A n k is the set of solutions of the system of equations {f 1 = 0,...,f s = 0}. Denote by k[x] := k[x 1,...,X n ] / I the coordinate ring of X. We have a contravariant functor (affine varieties) X k[x]. (noetherian k algebras) Principle 1.2. Everything we want to know about X is contained in its coordinate ring k[x]. Definition 1.3. For a noetherian ring R we denote by the maximal spectrum of R. Max(R) = {maximal ideals in R} As a first incarnation of our principle we have: Theorem 1.4 (Hilbert Nullstellensatz). Let X be an affine variety. There is a one to one correspondence {points in X} 1:1 Max(k[X]). Given a point p X, the corresponding maximal ideal m p = {f k[x] f(p) = 0} gives rise to the short exact sequence: 0 m p k[x] k 0 f f(p). Let T p X := (m p / m 2 p ) denote the tangent space of X at the point p. Definition 1.5. Let A be a noetherian domain, and let p A be a prime ideal. The ring { a } A p := b a A, b A\p is called the localization of A at p. Definition 1.6. An affine variety is a smooth curve if for all points p X the tangent space T p X is 1-dimensional. One can characterize smooth affine curves in the following way: Lemma 1.7. For an affine variety X the following conditions are equivalent: (i) X is a smooth curve (ii) k[x] is a Dedekind domain (iii) For all m Max(k[X]) the localization k[x] m is a principal ideal domain.

3 COHERENT SHEAVES ON AN ELLIPTIC CURVES 3 Proof. The equivalence (ii) (iii) is just a characterization of a Dedekind domain [AM69, Proposition 9.2, Theorem 9.3]. For the other equivalence, since (T p X) / = m p m 2 p is one dimensional, this vector space is generated by the image of an element in m p, say a. By Nakayama s Lemma [AM69, Proposition 2.8], the ideal m p is generated by a in k[x] mp. Definition 1.8. Let X be an affine curve. The category of coherent sheaves Coh(X) on X is the category k[x]-mod of noetherian k[x]-modules. 2. Modules over Dedekind domains Let A be any Dedekind domain, for example the coordinate ring k[x] = k[x,y] / (y 2 x 3 +x) of the curve X = V (y 2 x 3 +x) or the ring of Gaussian integers Z[i]. We shall investigate the module category of A which in the first example of a category of coherent sheaves. Let M be a noetherian A-module. Definition 2.1. The torsion part of M is defined as T(M) = {m M a A \ {0} am = 0}. There is a short exact sequence: 0 T(M) M M / T(M) 0 ( ) where T(M) and hence M / T(M) are noetherian and M / T(M) is by construction torsion free. Proposition 2.2. Every noetherian module M over A splits into a torsion part and a projective part: M = T(M) M / T(M). Proof. Let m Max(A) be a maximal ideal. The module (M / T(M)) m is torsion free. By the classification of modules over principal ideal domains, we know that (M / T(M)) m is already free. Therefore Ext 1 A (M/ T(M),T(M)) m = Ext 1 A m ((M / T(M)) m,t(m) m ) = 0. Since Ext 1 A (M/ T(M),T(M)) m = 0 for all maximal ideals we conclude that Ext 1 A (M/ T(M),T(M)) = 0. Therefore the short exact sequence ( ) splits. We collect some facts about the module category of A in the following: Proposition 2.3. Let A be a Dedekind domain. (i) The category A-mod is hereditary. (ii) If an A-module M is torsion free then there are ideals I 1,...,I s A such that M s = I i. (iii) If I and J are ideals in A then i=1 I J = IJ A.

4 4 KRISTIAN BRÜNING, IGOR BURBAN (iv) If M is a noetherian A-module then there exist ideals I A, m 1,..., m s Max(A) and integers m,p 1,...,p s Z >0, such that M s = I A m ( A / m p i i ). Definition 2.4. A k-linear category is called finite dimensional if all the Hom-spaces are finite dimensional as k-vector spaces. Remark 2.5. Since End(A) = A the category of A-modules is infinite dimensional. The assertion (i) can be shown by a local-global argument similar to the proof of 2.2. The statement (ii) tells us that the indecomposable projective modules are ideals. The category A-mod is not a Krull- Schmidt category due to (iii). Exercise 2.6. Let X be the cuspidal curve, i.e. X = V (y 2 x 3 ) and A := k[x,y] / (y 2 x 3 ). i=1 Show that the category A-mod is not hereditary. (Hint: find a module M with Ext 2 (M,M) 0) Construct an indecomposable A-module which is neither torsion nor torsion free. 3. Coherent sheaves on projective varieties In this section we shall introduce the category of coherent sheaves on a projective variety X and define some basic functors like the internal tensor product and the internal Hom-functor. Definition 3.1. The n-dimensional projective space over the field k is defined as P n k := kn+1 \ {0} / k, where two points x and x in k n+1 \ {0} are identified if there is a λ k such that x = λx. For an x = (x 0,...,x n ) k n+1 \ {0} let (x 0 : : x n ) denote its equivalence class which is called homogeneous coordinates. If f is a homogeneous polynomial of degree d, i.e. f(λ(x 0,...,x n )) = λ d f(x 0,...,x n ) for all λ 0, then f = 0 defines a subset of P n k. Recall that the radical I of an ideal I is {f k[x] n N : f n I}. Definition 3.2. Let I k[x 0,...,X n ] be a homogeneous ideal, that is generated by homogeneous polynomials f 0,...,f s such that I (X 0,...,X n ) then the corresponding projective variety X = PV (I) P n k

5 COHERENT SHEAVES ON AN ELLIPTIC CURVES 5 is the set of solutions of the system of equations {f 0 = 0,...,f n = 0}. Remark 3.3. The ideal X 0,...,X n does not define a projective variety and therefore is called the irrelevant ideal. Like in the affine case let k[x] = k[x 0,...,X n ] / I denote the coordinate ring of the projective variety X = PV (I). Observe that R = k[x] is a graded ring: R = R i R i R j R i+j, i=0 where R i is generated by the classes of homogeneous polynomials of degree i. Definition 3.4. Let grmod(r) be the category of finitely generated R- modules M = i=n 0 M i satisfying R i M j M i+j. Exercise 3.5. The category grmod(r) is finite dimensional. For a graded ring R let grmax(r) = {homogeneous prime ideals of coheight 1 in R} be its graded maximal spectrum. Here, a homogeneous prime ideal p has coheight 1 if the only proper homogeneous prime ideal m containing p is the irrelevant ideal. In analogy with the affine case we have a projective Hilbert Nullstellensatz: Theorem 3.6 (Hilbert Nullstellensatz). Let X be a projective variety. There is a one to one correspondence {points in X} 1:1 grmax(k[x]). The category grmod(k[x]) has less nice properties than its affine analogon k[x]-mod. So what are the problems with grmod(k[x])? If two projective varieties are isomorphic X = Y then k[x] = k[y ] in general. Example 3.7. The conic X 2 0 +X2 1 +X2 2 = 0 in P2 k is isomorphic to P1 k. But k[x 0,X 1,X 2 ] / (X X2 1 + X2 2 ) = k[y 0,Y 1 ]. The category grmod(k[p 1 k ]) = grmod(k[x 0,X 1 ]) is not hereditary. Definition 3.8. Let R be a graded ring. For M grmod(r), n Z define the shift M(n) of M via M(n) i = M(n + i). For S = k[y 0,Y 1 ] the sequence Y1 Y 0 S( 2) 0 S( 1) 2 ( Y 0 Y 1 ) S k 0 is a minimal projective resolution of k. Hence, gl. dim(grmod(s)) 2. It is actually equal to 2 due to Hilbert s Syzygy Theorem. Even worse: gl.dim(grmod(k[x 0,X 1,X 2 ] / (X0 2 +X2 1 +X2 2 ))) =. So we need to modify grmod(k[x]) to get a nice category worth to call Coh(X). Let X P n k be a projective variety with coordinate ring R = k[x] which is graded. Let A := grmod(r) denote the category of finitely generated

6 6 KRISTIAN BRÜNING, IGOR BURBAN graded R-modules and B := grmod 0 (R) the category of finite dimensional graded R-modules. In order to construct the quotient A / B the following is fact useful: Observation 3.9. Let 0 M N K 0 be exact in A. Then N B if and only if M and K are in B. Definition Let S B := {f Mor(A) ker(f),coker(f) B}. Exercise The set S B is multiplicatively closed (i.e. if f,g S B are composable morphisms then the composition f g is in S B and S B contains the identity morphisms). Having a multiplicatively closed set, we can form the localization with respect to that set: define the Serre quotient A / B := A[S 1 B ]. We shall give a second definition of the category A / B which has the advantage of a more concrete description of its morphisms. Let A and B be as above. The objects of the Serre quotient A / B are just the objects of A. For M, N A introduce the set: I M,N := {((X,ϕ),(Y,ψ)) X ϕ M, N ψ Y, coker(ϕ),ker(ψ) B}, and notice that, given ϕ and ψ, there is a map Hom A (M,N) Hom A (X,Y ). We endow the set I M,N with an ordering : ((X,ϕ),(Y,ψ))) ((X,ϕ ),(Y,ψ ))) if there are maps f and g in A such that the following diagrams commute: X f X ϕ ϕ N ψ M Y g Exercise Show that the partially ordered set I M,N is directed, that is for all a,b I M,N there is a c I M,N such that a c and b c. Now we can define the morphisms in our category A / B as the direct limit of the inductive system {Hom A (X,Y )} ((X,ϕ),(Y,ψ)) IM,N [AM69, 2, Exercise 14]: Hom (M,N) := lim Hom A /B A (X,Y ). I M,N ψ Y.

7 COHERENT SHEAVES ON AN ELLIPTIC CURVES 7 The effect of this definition is that a morphism M N in A / B is represented by a map X Y with smaller source and target. In a picture: X Y X Y making source and target smaller M N With this definition we have [Gab62]: Theorem 3.13 (Gabriel). The category A / B is abelian and the canonical functor A A / B is exact. For M A let M denote its image in the Serre quotient. Now we can give the definition of coherent sheaves in the projective setup: Definition The category of coherent sheaves on a projective variety X P n k is defined as the Serre quotient: Coh(X) := grmod(k[x]) / grmod 0 (k[x]). Remark The definition 3.14 of the category of coherent sheaves on a projective variety X is due to Serre [Ser55] and is different from the standard one, see e.g. [Har77]. This way to define Coh(X) via quotient categories does not require any special knowledges of the sheaf theory and plays a key role in the computer algebra approach to the study of coherent sheaves on projective varieties. This category has nice properties like in the affine case Theorem 3.16 (Serre). [Ser55] Let X P n k be a projective variety. The category Coh(X) is a finite dimensional, abelian, noetherian category. If X is smooth of dimension n then gl. dim(coh(x)) = n. Note that if X is a smooth projective curve then Coh(X) is hereditary. The category Coh(X) is Krull-Schmidt, because it is finite dimensional and abelian. Remark This theorem shows an important difference between projective and affine varieties: the category of coherent sheaves is always finite dimensional if the variety is projective but for an affine variety this is the case if and only if it has Krull dimension 0. In the case X = P 1 k we can classify the indecomposable objects. Example Let S = k[y 0,Y 1 ] with k algebraically closed. Then the indecomposable objects of Coh(P 1 k ) are O P 1 k (n) := S(n), n Z

8 8 KRISTIAN BRÜNING, IGOR BURBAN S / p s, where p = λy 0 µy 1, (λ : µ) P 1 and s Z >0. Compare the indecomposable objects of Coh(P 1 k ) and k[t]-mod. Functors on Coh(X). In the following we shall introduce the internal Homfunctor and the internal tensor product on the category of Coh(X), that is for M Coh(X) we shall define the functors M OX : Coh(X) Coh(X) and Hom( M, ) : Coh(X) Coh(X) which will be induced by the tensor product and Hom-functor on the category of graded modules over the coordinate ring k[x]. So let R = k[x], A = grmod(r) the category of finitely generated graded R-modules and B = grmod 0 (R) the category of finite dimensional graded R-modules. If M is in A then there is a functor M R : A A. Lemma The tensor functor M R maps the set S B of morphisms in A whose kernel and cokernel belong to B into itself. Proof. Let f : N K be a morphism of graded modules such that ker(f) and coker(f) are finite dimensional. We shall show that then for every finitely generated module M the modules ker(1 M f) and coker(1 M f) are finite dimensional. The functor M is right exact, so coker(1 M f) = M coker(f) is finite dimensional since the tensor product of a finite-dimensional module with a noetherian module is always finite dimensional. Consider the following commutative diagram with exact rows ker(a) M ker(f) t a M N 1M f M im(f) = 0 ker(1 M f) M N 1M f M K M coker(f). Since M ker(f) is finite dimensional, the submodule ker(a) is finite dimensional, too. The map s enters in the short exact sequence: Tor 1 (M,coker(f)) M im(f) s M K M coker(f) 0. Note that a graded noetherian R-module L is finite dimensional if and only if for all m grmax(r) it holds L m = 0. Since the localization functor is exact and commutes with tensor products, it also commutes with Tor i and we may conclude that Tor 1 (M,coker(f)) is finite dimensional. The canonical functor grmod(r) grmod(r) / grmod 0 (R) is exact by 3.13, so we know that s : M im(f) M K is an isomorphism in grmod(r) / grmod 0 (R). Projecting the commutative diagram above to grmod(r) / grmod 0 (R) and using the 5-lemma, we conclude that t is an isomorphism in the Serre quotient. By [Gab62] it means that ker(t) and coker(t) are finite dimensional and hence ker(1 M f) is finite dimensional, too. s 0

9 COHERENT SHEAVES ON AN ELLIPTIC CURVES 9 Using the universal property of the Serre quotient A / B = A[S 1 B ] we obtain: Corollary There is a unique endofunctor M OX on Coh(X) making the following diagram commutative grmod(r) M R grmod(r) Coh(X) Coh(X). fm OX In a similar way, for M A the functor gr Hom(,M) : A op A N n Z Hom(N,M(n)) induces Hom(, M) : Coh(X) op Coh(X). Analogously one can construct Hom( M, ). Remark The functor Hom( M, ) : Coh(X) op Coh(X) should not be mixed with the categorical Hom-functor Hom Coh(X) ( M, ) : Coh(X) Vect k. At the end of this section on coherent sheaves on projective varieties, we take a closer look at localization and an important class of coherent sheaves, namely the locally free ones and discuss some examples. Definition Let R be a graded domain and m grmax(r) a maximal ideal. Define the localization { } f R m := f R,g R \ m, deg(f) = deg(g). g The localization R m is a local non-graded ring. We have a functor which induces a functor grmod(r) mod(r m ) Coh(X) mod(r m ) F F m. Let m x grmax(r) be the maximal ideal corresponding to the point x X according to the Hilbert Nullstellensatz 3.6. Definition Let F be a coherent sheaf over a projective variety X. If x X then the localized sheaf F mx =: F x is called the stalk of F at x. The sheaf F is called locally free if for all x X the stalk F x is a free R mx -module.

10 10 KRISTIAN BRÜNING, IGOR BURBAN Examples of coherent sheaves. Let X be a projective curve, X P n k and R = R X = k[x] its coordinate ring. Remember, that for an R-module M the image under grmod(k[x]) grmod(k[x]) / grmod 0 (k[x]) = Coh(X) is denoted by M. There are examples of coherent sheaves: The structure sheaf O X = O = R Let x X and m x be the corresponding maximal ideal in R. The structure sheaf at a point x is k(x) := R / m x. The ideal sheaf of x is defined as J x := m x. We have a short exact sequence in Coh(X) 0 J x O X k(x) 0. If X is a smooth curve then O( x) := J x is locally free. The sheaf O(x) = Hom(O( x), O X ) occurs as the middle term in a short exact sequence 0 O O(x) k(x) Coherent sheaves on an elliptic curve In this section we shall restrict ourselves to a special class of projective varieties called elliptic curves, and describe all indecomposable coherent sheaves on them. So let X P n k be a projective variety, R = k[x] its coordinate ring and Coh(X) the category of coherent sheaves over X. We know that Coh(X) is abelian, noetherian, finite dimensional Krull-Schmidt category. So in order to classify the coherent sheaves we can confine ourselves to a description of the indecomposable ones. If X is in addition a smooth curve we know by 3.16 that Coh(X) is hereditary. The Calabi-Yau property of Coh(X), if X is an elliptic curve will be essential for the classification of indecomposable objects. Definition 4.1. An elliptic curve over a field k is a smooth plain projective curve of degree 3 given by a homogeneous polynomial and having a solution over k. P 3 (x,y,z) = 0 P 2 k We have seen examples of elliptic curves given by the equation zy 2 = 4x 3 axz 2 bz 3, where := a 3 27b 2 0, which looks in the affine chart z 0 like:

11 COHERENT SHEAVES ON AN ELLIPTIC CURVES 11 Calabi-Yau property. If X is an elliptic curve, then there is an important isomorphism between the Hom and the Ext functors: let F and G be coherent sheaves over X, then there is a bifunctorial isomorphism Hom(F, G) = Ext 1 (G, F). A hereditary abelian category A satisfying the property above is sometimes called Calabi-Yau category of dimension 1. There is a classification result by Reiten and van den Bergh [RVdB02, Theorem C] that characterizes hereditary finite dimensional noetherian abelian categories with Serre-duality. In our situation it reduces to: Theorem 4.2 (Reiten, van den Bergh). Let k be an algebraically closed field and A an abelian, noetherian, hereditary, indecomposable Calabi-Yau category with finite dimensional Hom-spaces. Then A is equivalent to the category of nilpotent representations of the quiver with one vertex and one loop: A Nil( ) or A is equivalent to the category of coherent sheaves on an elliptic curve X: A Coh(X). Let us now introduce some invariants of coherent sheaves on an elliptic curve that play a key role in the classification. Definition 4.3. For E and F in Coh(X) the Euler form of E and F is defined as E, F := dim k Hom(E, F) dim k Ext 1 (E, F). The Euler characteristic of F is by the definition χ(f) := O, F. We have the following properties: The Euler characteristic is additive in the following sense: if 0 F F F 0 is exact then: χ(f) = χ(f ) + χ(f ). Let F Coh(X), x X and m x grmax(r) then the following functors are exact Coh(X) R mx -mod Vect Quot(Rmx ) F F x F x Rmx Quot(R mx ). Definition 4.4. The rank of a coherent sheaf F is defined as rk(f) := dim Quot(Rmx )(F x Rmx Quot(R mx )). The following lemma assures that the rank is well-defined. Lemma 4.5. The rank of a coherent sheaf does not depend on the choice of the point x X.

12 12 KRISTIAN BRÜNING, IGOR BURBAN The rank is also additive with respect to short exact sequences (compare with the Euler characteristic). In fact we get (χ,rk) : K 0 (Coh(X)) Z 2, where K 0 denotes the Grothendieck group. The quotient of the Euler characteristic by the rank will be useful in the classification of the coherent sheaves. Definition 4.6. The slope of a coherent sheaf F is an element in Q defined as µ(f) = χ(f) rk(f). Lemma 4.7. If 0 F F F 0 is exact then exactly one of the following assertions is true: µ(f ) < µ(f) < µ(f ) µ(f ) > µ(f) > µ(f ) µ(f ) = µ(f) = µ(f ). Definition 4.8. A coherent sheaf F is called stable (or semi-stable) if for any non-trivial exact sequence 0 F F F 0 holds µ(f ) < µ(f) (µ(f ) µ(f)). Proposition 4.9. Let X be a smooth projective curve over a field k. If two semi-stable coherent sheaves E, F Coh(X) satisfy µ(e) > µ(f) then Hom(E, F) = 0. If F is stable then End(F) = K for a finite field extension k K. Let µ Q then the full subcategory SS µ X = {semi-stable coherent sheaves of slope µ} is abelian and artinian. Proof. We only show the first assertion: let E and F be coherent sheaves such that µ(e) > µ(f). Let f : E F be a morphism which is not zero. There are exact sequences E im(f) 0 0 im(f) F. By 4.7 we have µ(e) µ(im(f)) µ(f) which contradicts the assumption. The following theorem will be used to derive that all indecomposable coherent sheaves on an elliptic curve are semi-stable. Theorem 4.10 (Harder-Narasimhan, Rudakov). [HN75] Let X be a projective curve, then for a given F Coh(X) there is a unique filtration: F = F 0 F 1 F n F n+1 = 0 such that A i := F i / Fi+1 for 0 i n are semi-stable and µ(a 0 ) < µ(a 1 ) < < µ(a n ).

13 COHERENT SHEAVES ON AN ELLIPTIC CURVES 13 In the sequel, we shall abbreviate the filtration of this theorem by HNF (Harder-Narasimhan-filtration) and call n the length of the filtration. Corollary Let X be an elliptic curve and F Coh(X) indecomposable. Then F is semi-stable. Proof. Assume F is indecomposable but not semi-stable. If the HNF has length 1 we get a short exact sequence 0 F 1 F A 0 0 such that F 1 and A 0 are semi-stable. By the second property of the HNF we know that µ(a 0 ) < µ(f 1 ). The Calabi-Yau property of Coh(X) yields that Ext 1 (A 0, F 1 ) = Hom(F 1, A 0 ). By 4.9 we know that Hom(F 1, A 0 ) = 0. Hence Ext 1 (A 0, F 1 ) = 0 and the short exact sequence above splits. So, F = F 1 A 0 which is a contradiction. The general argument is an induction on the length of the HNF. Corollary If X is an elliptic curve then Coh(X) = SS µ X. µ Q χ Ext SS µ µ<µ Hom SS µ To sum up, we can decompose every coherent sheaf on an elliptic curve into a direct sum of semi-stable ones. So the classification boils down to a description of semi-stable sheaves. We shall see that SS µ X = SS µ X = SS X µ,µ Q, where SSX is the category of torsion sheaves. In order to prove this fact we have to introduce some preliminaries. Theorem 4.13 (Riemann-Roch formula). Let X be an elliptic curve and E and F coherent sheaves over X. Then E, F = χ(f)rk(e) χ(e)rk(f). Remark By the Calabi-Yau property we have E, F = F, E. rk

14 14 KRISTIAN BRÜNING, IGOR BURBAN Recall that the left radical (respectively right radical) is the set l.rad, = {F K 0 (Coh(X)) F, = 0} (respectively r.rad, = {F K 0 (Coh(X)), F = 0}). Corollary From the Riemann-Roch formula it follows that the right and the left radical coincide and are both equal to: l.rad, = r.rad, = {F K 0 (Coh(X)) rk(f) = χ(f) = 0}. Example Let x 1,x 2 X. Then 0 [k(x 1 )] [k(x 2 )] l.rad,. If the left and the right radical of a bilinear form coincide we shall call it just the radical. As a consequence of the Corollary 4.15 we get an isomorphism Z := ( rk χ ) : K0 (Coh(X)) / rad, Z2. The charge of a sheaf E is defined to be Z(E). Remark The map Z is surjective since Z(O X ) = ( 1 0 ), Z(k(x)) = ( 0 1 ). The group of exact auto-equivalences Aut(D b (Coh(X))) of the bounded derived category of coherent sheaves on X acts on K 0 (Coh(X)) by automorphisms, since K 0 is actually an invariant of the derived category. This action preserves the Euler form and its radical. Hence, we get a group homomorphism π : Aut(D b (Coh(X))) SL 2 (Z) which sends an auto-equivalence f to the upper map in the following commutative diagram: Z 2 Z 1 = Z 2 = Z K 0 (Coh(X)) / rad, f = K 0 (Coh(X)) / rad, which is in SL 2 (Z). In the sequel we show that π is surjective by defining two derived equivalences A and B which are mapped under π to the matrices A := ( ) respectively B := ( ). There is an evaluation map in D b (Coh(X)) ev F : RHom(E, F) L E F. Definition For a coherent sheaf E on a projective variety X the twist functor is defined as T E : D b (Coh(X)) D b (Coh(X)) F cone(ev F )

15 COHERENT SHEAVES ON AN ELLIPTIC CURVES 15 Remark If E and F are coherent sheaves on a projective curve X such that Ext 1 (E, F) = 0 then T E (F) is isomorphic to the complex 0 Hom(E, F) k E ev F F 0. In particular, if ev F is injective then T E (F) = coker(ev F ) and if ev F is surjective then T E (F) = ker(ev F )[1] The following theorem shows that this notion is useful: Theorem 4.20 (Seidel-Thomas). [ST01, 2b] The assignment T E is a triangle functor. By definition there is a triangle RHom(E, F) L E ev F T E (F) Σ(RHom(E, F) L E). Hence in K 0 (Coh(X)) it holds: T E : [F] [F] E, F [E]. The following theorem says when a twist functor is an equivalence. Theorem 4.21 (Seidel-Thomas, Lenzing-Meltzer). [ST01, Proposition 2.10] [LM00] Let X be an elliptic curve. If a coherent sheaf E is endo-simple, i.e. End(E) = k, then the twist functor T E is an exact equivalence of D b (Coh(X)). Example The sheaves O X and k(x) for a k-point x X are endosimple. Lemma [ST01, formula 3.11] There is an isomorphism of functors: T k(x) = O(x). So by the Theorem 4.21 we know that A := T O and B := T k(x) are autoequivalences. Let A := π(a) and B := π(b) be the images in SL 2 (Z). Now we shall determine the matrices A and B by looking at the action of maps A and B in the Grothendieck group: and On the other hand T O : [O] [O] O, O [O] = [O] T O : [k(x)] [k(x)] O,k(x) [O] = [k(x)] [O]. T k(x) : [k(x)] [k(x)] k(x),k(x) [k(x)] = [k(x)] and T k(x) : [O] [O] k(x), O [k(x)] = [O] + [k(x)]. Since Z(O X ) = ( 1 0 ) and Z(k(x)) = ( 0 1 ), it follows that A = ( ) and B = ( ). Since A and B generate SL 2(Z), we conclude that is surjective. π : Aut(D b (Coh(X))) SL 2 (Z)

16 16 KRISTIAN BRÜNING, IGOR BURBAN Classification of the indecomposable coherent sheaves. Now we are able to classify the indecomposables in Coh(X). Let F be indecomposable in Coh(X) of charge Z(F) = ( d r ) Z 2 and if d 0 let n = gcd(r,d) be the greatest common divisor. If d 0 then there is a matrix F SL 2 (Z) with F ( d r ) = ( n 0 ). If d = 0 let F be the matrix ( ) in SL2 (Z) which flips the coordinate axes. Since π : Aut(D b (Coh(X))) SL 2 (Z) is surjective we can lift F to an autoequivalence F Aut(D b (Coh(X))). Since F is an equivalence of categories the image F(F) is indecomposable again. The object F(F) is then a shift of a coherent sheaf of rank 0 because Coh(X) is hereditary. Therefore F(F) is a torsion sheaf T and F = F 1 (T ). It means that any indecomposable coherent sheaf can be obtained from a torsion sheaf by applying an exact auto-equivalence of the derived category D b (Coh(X)). The indecomposable torsion sheaves are known: if m grmax(r) and s > 0 then So we have: F = F 1 ( R X /m s ). Theorem 4.24 (Atiyah). [Ati57, Theorem 7] If X P 2 k is an elliptic curve then an indecomposable coherent sheaf on X is uniquely determined by its charge in Z 2 and one continuous parameter m grmax(r). Corollary The category Coh(X) is tame. Summary. Let X be an elliptic curve. ( The indecomposable objects of Coh(X) are semi-stable: Coh(X) = SS µ X where SS µ X = {semi-stable sheaves of slope µ}. For µ,µ Q { } there is an equivalence of abelian categories SS µ X = SS µ X induced by an auto-equivalence of Db (Coh(X)). An indecomposable object in Coh(X) is determined by its charge and a maximal ideal m grmax(r). Let F and F be indecomposable such that Z(F) = (r,χ) and Z(F ) = (r,χ ). If χ r > χ r then Hom(F, F ) = 0 and dim Ext 1 (F, F ) = χr χ r. If χ r < χ r then dim Hom(F, F ) = χ r χr and Ext 1 (F, F ) = 0. In the rest of these notes we shall investigate the group Aut(D b (Coh(X))) and indicate that it is closely related to the braid group B 3 acting on 3 strands. At the end we shall cite Burban s and Schiffmann s theorem on the structure of the Hall algebra of Coh(X) which uses the braid group action on D b (Coh(X))). Recall that A := T O and B = T k(x) and their images under π are A := ) respectively B := ( ). Note the following equalities: ABA = BAB; (AB) 3 = ( ).

17 COHERENT SHEAVES ON AN ELLIPTIC CURVES 17 It turns out that these relations can be lifted to the group Aut(D b (Coh(X))): Theorem 4.26 (Mukai, Seidel-Thomas). [ST01, Proposition 2.13] ABA = BAB (AB) 3 = i [1] for an involution i : X X. This theorem implies that the group A, B generated by A and B is isomorphic to SL 2 (Z) which is a central extension of SL 2 (Z): 0 Z SL 2 (Z) SL 2 (Z) 1. We shall illustrate Theorem 4.26 by showing ABA(O) = BAB(O). Recall that the twist functor T E fits by definition in the triangle: RHom(E, F) L E ev F T E (F) Σ(RHom(E, F) L E). ( ) To compute the action of ABA on O we have to compute the triangle ( ). The functor A maps O to itself since the triangle ( ) for E = F = O is isomorphic to O O[ 1] (id w ) O O + where w Ext 1 (O, O). The functor B is given by tensoring with the sheaf O(x) by Hence, B(O) = O(x). Since there is a short exact sequence 0 O O(x) k(x) 0 and Ext 1 (O, O(x)) = 0 we find that A(O(x)) = k(x). So we have shown: Similar computations show that Hence ABA(O) = BAB(O). O A O B O(x) A k(x). O B O(x) A k(x) B k(x). Remark Some versions of twist functors appeared in the literature under different names: name appearance shrinking functors Ringel, tubular algebras [Rin84] reflection functors Mukai, K3-surfaces tubular mutations Lenzing-Meltzer [LM00] monodromy transformations mirror symmetry Fourier-Mukai transformations algebraic geometry Let X be an elliptic curve over a finite field k = F q. As we have seen, the category of coherent sheaves Coh(X) is a finite dimensional abelian hereditary category over k. By a general construction of Ringel [Rin90] one can attach to X an associative algebra H(Coh(X)) over the field K = Q( q). As a K- vector space H(Coh(X)) = K[F]. F Iso(Coh(X))

18 18 KRISTIAN BRÜNING, IGOR BURBAN and for F,G Iso(Coh(X)) the product is defined as Hom(F, G) [F] [G] = Ext 1 PF,G H [H], (F,G) H Iso(ρA) where P H F,G = 1 a F a G {0 G H F 0} is the orbifold number of short exact sequences with end terms F and G and the middle term H, and a X denotes the order of the automorphism ring of the object X. However, for applications it is more convenient to consider the extended Hall algebra H(Coh(X) := H(Coh(X)) K[Z 2 ] where Z 2 = K(X)/rad,. The main reason to enlarge the Hall algebra H(Coh(X)) to H(Coh(X)) is that due to a work of Green [Gre94] the algebra H(Coh(X)) carries a natural bialgebra structure. It turns out that our approach to coherent sheaves on elliptic curves via derived categories gives a proper tool to study properties of the extended Hall algebra H(Coh(X)). Theorem 4.28 (Burban-Schiffmann). [BS] Let X be an elliptic curve over the field k = F q. Then the group Aut(D b (Coh(X))) acts on the reduced Drinfeld double of the extended Hall algebra H(Coh(X)) by algebra homomorphisms. This symmetry allows to construct a certain natural subalgebra U(X) D H(Coh(X)) and show that U(X) is a flat deformation of the ring Q( q)[s ±1,t ±1 ][x ±1 1,...,x±1 n,...,y±1 1,...,y±1 m,...]sym of symmetric Laurent series. References [AM69] M. F. Atiyah and I. G. Macdonald. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., [Ati57] M. F. Atiyah. Vector bundles over an elliptic curve. Proc. London Math. Soc. (3), 7: , [BS] Igor Burban and Olivier Schiffmann. On the Hall algebra of an elliptic curve, I. arxiv:math.ag/ [Gab62] Pierre Gabriel. Des catégories abéliennes. Bull. Soc. Math. France, 90: , [Gre94] J. A. Green. Hall algebras and quantum groups. Textos de Matemática. Série B [Texts in Mathematics. Series B], 4. Universidade de Coimbra Departamento de Matemática, Coimbra, [Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, Graduate Texts in Mathematics, No. 52. [HN75] G. Harder and M. S. Narasimhan. On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann., 212: , 1974/75. [LM00] Helmut Lenzing and Hagen Meltzer. The automorphism group of the derived category for a weighted projective line. Comm. Algebra, 28(4): , [Rin84] Claus Michael Ringel. Tame algebras and integral quadratic forms, volume 1099 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, [Rin90] Claus Michael Ringel. Hall algebras and quantum groups. Invent. Math., 101(3): , 1990.

19 COHERENT SHEAVES ON AN ELLIPTIC CURVES 19 [RVdB02] I. Reiten and M. Van den Bergh. Noetherian hereditary abelian categories satisfying Serre duality. J. Amer. Math. Soc., 15(2): (electronic), [Ser55] Jean-Pierre Serre. Faisceaux algébriques cohérents. Ann. of Math. (2), 61: , [ST01] Paul Seidel and Richard Thomas. Braid group actions on derived categories of coherent sheaves. Duke Math. J., 108(1):37 108, [Ste75] Bo Stenström. Rings of quotients. Springer-Verlag, New York, Die Grundlehren der Mathematischen Wissenschaften, Band 217, An introduction to methods of ring theory. Kristian Brüning, Institut für Mathematik, Universität Paderborn, Paderborn, Germany. address: Igor Burban, Max-Planck-Institut für Mathematik, Bonn, Germany. address: